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Chapter 1

CONSTRUCTIONS AND CHARACTERIZATIONS OFCLASSICAL SETS IN PG(n,q)

Frank De Clerck and Nicola Durante ∗

Ghent University and Università di Napoli

Abstract

In this chapter we are interested in characterization theorems of the point sets ofclassical objects such as conics, quadrics, Hermitian varieties and (Baer) subgeome-tries in terms of their intersection with respect to subspaces. We will give some con-structions of sets that have the same type of intersection with subspaces as the classicalexample. first revision, december 28th

Key Words: Quadrics, Conics, Maximal arcs, Hermitian Varieties, Subspaces.

AMS Subject Classification: 51E, 05B.

1 Introduction

A quadric (sometimes called a hyperquadric or quadratic variety) Q in PG(n,q) is a varietythat can be described by a quadratic form

Q(x0,x1, . . . ,xn) =n

∑i, j=0

ai, jxix j.

If q is odd and the quadric is non-singular, the points on the quadric can be regarded asthe set of absolute points of an orthogonal polarity.

Quadrics in PG(2,q) are called conics.A Hermitian variety of PG(n,q2) is the set of absolute points of a non-degenerate uni-

tary polarity. If n = 2, it is called a Hermitian curve.A subgeometry of order pt of PG(n,q), q = ph, t|h, is the projective subgeometry in-

duced by a subset of points of PG(n,q) whose coordinates, with respect to a suitable frame,

∗[email protected], [email protected]

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2 Frank De Clerck and Nicola Durante

are in GF(pt). If h is even, a subgeometry of order ph/2 of PG(n,q) is called a Baer subge-ometry; in particular a Baer subline if n = 1 and a Baer subplane if n = 2. We remark thatHermitian varieties of PG(1,q2) and Baer sublines are the same objects.

A point P of a subset K of the point set of PG(n,q) is a singular point of K if all lineson P intersect K either in 1 or in q+1 points. The point set K is singular if it has a singularpoint.

A set K of points of PG(n,q) is said to be a set of class [m1, . . . ,mk]r, 1 ≤ r ≤ n−1, iffor every r-dimensional subspace π, |π∩K | = mi, 1 ≤ i ≤ k. It is said to be a set of type(m1, . . . ,mk)r if every mi actually occurs for some r-dimensional space π.

In this chapter, our main interest will go to sets in PG(n,q) of class [m1, . . . ,mk]1 or ofclass [m1, . . . ,mk]n−1 for which the number of intersection numbers mi is small and suchthat there is a classical example known. Actually, if the dimension r of the intersectingsubspace is clear, we often will omit the index r in this notation.

Let K be any set of points in PG(n,q), and embed PG(n,q) as a hyperplane Σ∞ inPG(n + 1,q). The point-line geometry T ∗

n (K ), called the linear representation of K , isconstructed as follows: the points of the geometry T ∗

n (K ) are the points of the affine spaceAG(n + 1,q) = PG(n + 1,q) \ Σ∞ and the lines of the geometry T ∗

n (K ) are the lines ofPG(n+1,q) not in Σ∞ and which meet Σ∞ in a point of K . The point graph of the geometryT ∗

n (K ) is denoted by Γ(K ).Let K be a set of points in PG(n,q) which is of type (m1,m2)n−1, also called a two-

character set with characters m1,m2; then the following nice theorem is commonly known.

Theorem 1.1 (Delsarte, see [34]). Let K = {Pi : i = 1,2, . . . , |K |}, where each Pi is anelement of GF(q)n+1, be a two-character set in PG(n,q), with characters m1,m2. If Kgenerates PG(n,q), then

1. the graph Γ(K ) is a strongly regular graph;

2. the code {(x ·P1,x ·P2, . . . ,x ·P|K |) : x∈GF(q)n+1} is a linear two-weight [|K |,n+1]-code with weights |K |−m1, |K |−m2;

3. the set D = {v ∈ GF(q)n+1 : 〈v〉 ∈ K } is a {λ1,λ2}-difference set over GF(q), forsome {λ1,λ2}.

See [34] for a comprehensive survey of two-character sets, two-weight codes, {λ1,λ2}-difference sets and the related strongly regular graphs.

2 Classical sets with few intersection numbers in PG(2,q)

2.1 Conics, ovals and hyperovals

A k-arc K in PG(2,q) is a set of k points which is of class [0,1,2]. It is immediately clearthat |K | ≤ q + 2. A (q + 2)-arc is called a hyperoval and can only exist if q is even, anexample being a conic C together with its nucleus N, also called a regular hyperoval (orhyperconic). A (q + 1)-arc in PG(2,q) is called an oval. Assume q is even; take a regular

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Constructions and characterizations of classical sets in PG(n,q) 3

hyperoval C ∪{N} and delete any point P different from the nucleus N, then this yields anoval (C ∪{N})\{P} also called a pointed conic, which has canonical form

{(1, t,√

t) : t ∈ GF(q)}∪{(0,0,1)}.

A pointed conic cannot be a conic if q ≥ 8 as two different conics have at most 4 points incommon.

In a very well known theorem, Segre [95] proves however that every (q + 1)-arc inPG(2,q), q odd, is a conic. The method of proof of Segre’s theorem is ingenious. Wemay take three points of the oval to be P1 : (1,0,0), P2 : (0,1,0), and P3 : (0,0,1) and ifP(a0,a1,a2) is a further point on the oval and x1 = λ0x2, x2 = λ1x0, x0 = λ2x1 are the threesecants PP1,PP2,PP3, then immediately λ0λ1λ2 = 1. Since the product of all non-zeroelements in the field is −1, it will follow (known as The Lemma of Tangents) that for thetangents at P1,P2,P3 it is x1 = k0x2, x2 = k1x0, x0 = k2x1: k0k1k2 = −1. From this followsthat the inscribed triangle and its circumscribed triangle are perspective with respect to thecenter (1,k0k1,−k1). It follows generally that every inscribed triangle and its circumscribedtriangle are perspective. Using this relation on the triangles formed from P,P1,P2 and P3,the coordinates of P satisfy a quadratic equation which becomes x0x1 + x1x2 + x2x0 = 0 ofa conic C .

2.1.1 Known hyperovals

A hyperoval O in PG(2,q) (q = 2h,h > 1) contains at least 4 points, no three of which arecollinear. Without any restriction we may assume that O passes through the four points(1,0,0),(0,1,0),(0,0,1) and (1,1,1), which implies that it is completely determined by itsaffine points (x,y,1). We define y = f (x) if and only if (x,y,1) is a point of O. It is easilyseen that f (x) is a permutation polynomial which is called an o-polynomial.

A lot of the known examples can be described by an o-polynomial ofthe form f (x) = xk, also called a monomial o-polynomial. Define D(h) ={k‖xk is an o-polynomial over GF(2h)}.

Theorem 2.1. If k ∈ D(h) then 1/k,1− k,1/(1− k),k/(k− 1) and (k− 1)/k (all takenmodulo 2h−1) are also elements of D(h) and yield projectively equivalent hyperovals.

We give a short description of the known elements in D(h); the related hyperovals arecalled monomial hyperovals.

1. It is clear that 2∈D(h), for all h and gives the regular hyperoval. Actually it is knownthat if h ≤ 3, every hyperoval in PG(2,2h) is a regular hyperoval.

2. It was proved by Segre [96] that 2i ∈ D(h) if and only if gcd(i,h) = 1. They arecalled translation hyperovals since they admit as an automorphism group a group oftranslations which acts transitively on the affine points of the hyperoval. When i 6= 1,h− 1, these hyperovals are not equivalent to regular hyperovals and examples existfor h ≥ 5, but h 6= 6.

3. Another class of monomial hyperovals is given by f (x) = x6, in case h is odd. Thesehyperovals were also discovered by Segre [99] in 1962, see also [100] for more de-tails.

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4. Let σ and γ be automorphisms of GF(2h), h odd, such that γ4 ≡ σ2 ≡ 2 (mod 2h−1)then Glynn [64] proved that γ+σ and 3σ+4 are elements of D(h).

Remarks

1. Glynn has checked by computer the possible values for monomial o-polynomials,given h, and from his search follows that no other monomial o-polynomials exist forh ≤ 28.

2. There are several hyperovals known which are not of the monomial type, but it shouldtake us too far to go into this. We refer to the nice electronic overview of Cherowitzofor all updated information on hyperovals.

3. The smallest plane that can contain an irregular hyperoval, is the plane PG(2,16),which contains up to isomorphism exactly one irregular hyperoval, the Lunelli-Scehyperoval [81]. Its automorphism group has order 144 and is acting transitively onthe points of the hyperoval. For a more detailed discussion on this hyperoval, werefer again to Bill Cherowitzo’s hyperoval webpage, where also a full description ofthe 6 non-equivalent hyperovals in PG(2,32) is given.

4. For more details on the codes related to hyperovals we refer to Chapter 1

2.1.2 Characterization theorems of conics and related sets

The theorem by Segre stipulates that a set of size q+1 and of type (0,1,2) in a Desarguesianprojective plane PG(2,q), q odd, is a conic C . A point P in the plane not on the conic C iscalled an interior point of the conic if it is incident with no tangent to the conic, while it iscalled an external point of the conic if it is incident with exactly two tangents to the conic.A conic has 1

2 q(q + 1) external points and 12 q(q− 1) interior points. The polar line of an

external point with respect to the conic is a secant to the conic, while the polar line of aninterior point with respect to the conic is an exterior line to the conic.

The set of external points of a conic in PG(2,q), q odd, is clearly a set of type(12(q−1), 1

2(q+1),q). One can wonder whether this characterizes this set indeed. The

following theorem from 1983 gives the status of knowledge until 2007.

Theorem 2.2 ( [56]). If K is a set in PG(2,q), q odd, which is of type(1

2(q−1), 12(q+1),q

)and |K |< q(q+1)/2+q/5, then |K |= q(q+1)/2 and K is the set of external points of aconic.

In 2007, the theorem has been improved as follows.

Theorem 2.3 ( [38]). If K is a set in PG(2,q), q odd, which is of type(12(q−1), 1

2(q+1),q), then |K | = q(q + 1)/2 and K is the set of external points of a

conic.

The set of internal points with respect to a conic in PG(2,q), q odd, is a set of type(0, 1

2(q−1), 12(q+1)

). Several theorems, trying to characterize the set of internal points of

a conic have been proved in the 80’s. See for instance [1] and the following theorem.

http://www-math.cudenver.edu/~wcherowi/research/hyperoval/hypero.html

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Theorem 2.4 ( [55]). If K is a set in PG(2,q), q odd, which is of type(0, 1

2(q−1), 12(q+1)

), then (q2−2q−1)/2 < |K |< (q2 +1)/2.

The following theorem gives however the best characterization.

Theorem 2.5 ( [38]). If K is a set in PG(2,q), q odd, which is of type(0, 1

2(q−1), 12(q+1)

), then |K |= q(q−1)/2 and K is the set of interior points of a conic.

Remarks

It is not difficult to prove the following results as a corollary of the above theorems.

1. A set of type(1, 1

2(q+1), 12(q+3)

)in PG(2,q), q odd, is the union of a non-

degenerate conic C and its internal points.

2. A set of type(1

2(q+1), 12(q+3),q+1

)in PG(2,q), q odd, is the union of a non-

degenerate conic C and its external points.

3. Actually, in [38] all sets of class [0, 12(q− 1), 1

2(q + 1),q] in PG(2,q), q odd, areclassified, and so also their complements being sets of class [1, 1

2(q+1), 12(q+3),q+

1].

2.2 Maximal arcs

2.2.1 Introduction

A {k;d}-arc K , in a finite projective plane of order q, is a non-empty proper subset of kpoints such that some line meets K in d points, but no line meets K in more than d points.For given q and d, an easy counting argument shows that k≤ q(d−1)+d. If equality holds,K is called a maximal arc of degree d. A maximal arc K can be defined as a non-empty,proper subset of points of the projective plane such that every line meets the set in 0 or dpoints, for some d. Trivial examples are the following ones.

• Any point of a projective plane of order q is a maximal {1;1}-arc in that plane.

• An affine plane of order q in a projective plane of order q is a {q2;q}-arc.

For the remainder we will discard these trivial examples. A point of the plane not on themaximal arc K is incident with q + 1− q/d lines each intersecting K in d points. Hence,if K is a maximal arc of degree d, then d should divide q. Moreover, from this observationfollows that the set of external lines to the maximal arc (i.e. the lines not intersecting K )constitute a maximal arc K ′ of degree q/d in the dual projective plane. Hence, any maximalarc K of degree d in PG(2,q) yields another maximal arc K ′, also called the dual maximalarc of degree q/d, in PG(2,q). Examples of non-trivial maximal arcs in even characteristicplanes are known since 1969 (the Denniston maximal arcs, see further). It has been conjec-tured by several authors that non-trivial maximal arcs could not exist in PG(2,q), q odd. Ithas taken more than 25 years to prove this.

Theorem 2.6 ( [10]). No non-trivial maximal arcs exist in PG(2,q) when q is odd.

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For the polynomial technique used to prove this result and for a more recent and moregeneral theorem we refer to Chapter 2. Hence, from now on we may assume that q and dare powers of 2. In the next section we will describe the known examples and give somecharacterization theorems.

2.2.2 The known constructions of maximal arcs

We will describe the constructions of maximal arcs, known so far. The oldest constructionis due to Denniston [43], while the most recent construction is due to Mathon [82]. Bothconstructions are more algebraic while the two constructions of Thas [107, 108] are moregeometric. The Denniston maximal arcs are a special case of those of Mathon type, andhence we will start with the more general class being the maximal arcs of Mathon type.

The construction by R. Mathon

Let Tr be the usual absolute trace map from GF(q) onto GF(2). Represent the points ofPG(2,q) via homogeneous coordinates (a,b,c), the lines as triples [u,v,w] over GF(q) andthe incidence by the usual inner product au+bv+ cw = 0.

For α,β ∈ GF(q), with Tr(αβ) = 1, and λ ∈ GF(q), define Fα,β,λ to be the conic

Fα,β,λ = {(x,y,z) : αx2 + xy+βy2 +λz2 = 0}

and let F be the union of all such conics. All conics in F have the point F0 = (0,0,1) astheir nucleus.

For given λ 6= λ′, define a composition

Fα,β,λ⊕Fα′,β′,λ′ = Fα⊕α′,β⊕β′,λ⊕λ′

where the operator ⊕ is defined on GF(q)×GF(q) by

α⊕α′ =

αλ+α′λ′

λ+λ′, β⊕β

′ =βλ+β′λ′

λ+λ′, λ⊕λ

′ = λ+λ′.

Lemma 2.7 ( [82]). Two non-degenerate conics Fα,β,λ, Fα′,β′,λ′ , λ 6= λ′, and their composi-tion Fα,β,λ⊕Fα′,β′,λ′ are mutually disjoint if Tr((α⊕α′)(β⊕β′)) = 1.

Given some subset C of F , we say C is closed if for every Fα,β,λ 6= Fα′,β′,λ′ ∈ C ,Fα⊕α′,β⊕β′,λ+λ′ ∈ C .

Theorem 2.8 ( [82]). Let C be a closed set of conics with a common nucleus F0 in PG(2,q),q even. Then the union of the points of the conics of C together with F0 form a maximal arcof degree |C |+1 in PG(2,q).

For examples of maximal arcs of Mathon type and more information, we refer to [58],[59], [66], [67], [68], [82].

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The construction by R. Denniston

The maximal arcs of Denniston type are a special case of those of Mathon type. Chooseα∈GF(q) such that Tr(α) = 1. Let A be a subset of GF(q)? such that H = A∪{0} is closedunder addition. Then the set of conics {Fα,1,λ : λ∈ A} together with the nucleus F0 is the setof points of a maximal arc of degree |H| in PG(2,q), which yields exactly the constructionof Denniston [43]. Actually, it is known that the dual of a Denniston maximal arc is againof Denniston type (see for instance [69] for a proof).

The constructions by J. A. Thas

In 1974, Thas [107] gave the following construction of maximal arcs of degree q in trans-lation planes of order q2. We first quickly describe the so-called André-Bruck-Bose repre-sentation ( [8], [27], [28]) of these planes. Let PG(3,q) be embedded as a hyperplane Σ∞ inPG(4,q). Let S be a spread of Σ∞. Then the following incidence structure π of points andlines is an affine plane of order q2, known as a translation plane (with kernel containingGF(q)). The points of π are the points of PG(4,q) \Σ∞, the lines of π are the planes ofPG(4,q) meeting Σ∞ in a line of S ; incidence is the natural inclusion. The affine plane π

can be completed to a projective plane by adding the points at infinity represented by theelements of S . The projective plane is Desarguesian if and only if the spread S is regular(i.e. the regulus defined by any 3 lines of S is completely contained in S ).

The construction of Thas goes as follows. Let O be an ovoid and S a spread of Σ∞ suchthat each line of S has exactly one point in common with O. If X ∈ PG(4,q)\Σ∞, then theunion K of points on the lines joining X and O forms a maximal {q3−q2 +q;q}-arc in thetranslation plane π of order q2 defined by S . The known ovoids of PG(3,q), q even, arethe elliptic quadrics and the Tits ovoids defined for q = 22e+1, e ≥ 1 (see Section 3.1.3). IfS is the regular spread and the ovoid O is the elliptic quadric, then the maximal arc is ofDenniston type (see also [69]). If O is the Tits ovoid and S is the regular spread, then itis not of Denniston type neither of Mathon type. Penttila [69] found that there are exactlytwo orbits on Tits ovoids in the stabilizer of a regular spread, yielding two families of Thasmaximal arcs of degree q in PG(2,q2), q = 22e+1, e ≥ 1, associated with Tits ovoids.

In 1980, Thas [108] employed quadrics and spreads in projective spaces to constructdegree qt−1 maximal arcs in symplectic translation planes of order qt . Let Q − = Q −(2t−1,q) be a non-singular elliptic quadric and let S be a (t − 1)-spread in PG(2t − 1,q) ofwhich the restriction to Q − is a (t − 2)-spread. Embed PG(2t − 1,q) as a hyperplane Σ∞

in PG(2t,q). If X ∈ PG(2t,q) \Σ∞ then the union K of points on the lines joining X andQ − form a maximal {q2t−1−qt +qt−1;qt−1}-arc in the translation plane π of order q2(t−1)

defined by S . We note that for t = 2 we obtain the former construction of Thas in which theovoid is an elliptic quadric. If S is a spread such that the plane π is Desarguesian then K isa Denniston maximal arc, i.e. all the Thas maximal arcs of this type in Desarguesian planesare of Denniston type (see also [69]).

Remark

In [20] it has been proved that a (t−1)-spread S in PG(2t−1,q) of which the restriction toQ − is a (t−2)-spread cannot exist if q is odd, and hence the construction of Thas does not

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2.2.3 Some characterization theorems for maximal arcs

In this section we will give some characterization theorems that we think are important, butthis is of course not the complete list of all characterization theorems.

The following theorem due to V. Abatangelo and B. Larato gives up to our knowledgea first characterization of the maximal arcs of Denniston type that can be seen indeed as thepencil K =

Sλ∈H Fλ of conics Fλ = Fα,1,λ, H an additive subgroup of order d of GF(q),+;

q = 2h, Tr(α) = 1.

Theorem 2.9 ( [3]). 1. If K =S

λ∈H Fλ is a maximal arc for some subset H of GF(q),then H must be a subgroup of the additive group of GF(q).

2. If a maximal arc of PG(2,q), q even, is invariant under a cyclic linear collineationgroup of order q+1, then it is a Denniston arc.

Actually, the full stabilizer of a Denniston maximal arc has been completely describedby Hamilton and Penttila [69].

Theorem 2.10 ( [69]). Let K be a maximal arc of Denniston type in PG(2,2h), h > 2,which is of degree d, 2 < d < q/2. Let H be the additive subgroup of GF(q),+ of orderd defining the maximal arc. Define the group G acting on GF(2h) by G = {x 7→ axσ :a ∈ GF(2h)∗,σ ∈ AutGF(22h)}. Then the collineation stabilizer of K is isomorphic toC2h+1 o GH , the semidirect product of a cyclic group of order 2h + 1 with the stabilizer ofH in G.

As far as the Thas maximal arcs of degree q in PG(2,q2) are concerned, as alreadymentioned they are isomorphic to a Denniston maximal arc if the ovoid O is an ellipticquadric Q−(3,q). When the ovoid is the Tits ovoid, then it yields two non-isomorphicmaximal arcs which are not of Denniston type. The following theorem gives all informationon the stabilizer of these maximal arcs.

Theorem 2.11 ( [69]). There are, up to equivalence under PΓL(3,q2), q = 22e+1, e ≥ 1,two maximal arcs of Thas type in PG(2,q2) arising from Tits ovoids. They have stabilizersin PΓL(3,q2) given by the semidirect product of a dihedral group of order 4(q± (2q)

12 +

1)(q−1) by a cyclic group of order 2e+1.

Finally, the next theorem is also proved in [69].

Theorem 2.12 ( [69]). Let K be a non-trivial maximal arc in PG(2,q), q > 2, such thatthe collineation stabilizer of K acts transitively on the points of K , then K is isomorphicto one of the following.

1. A regular hyperoval in PG(2,2) or PG(2,4), or a Lunelli-Sce hyperoval in PG(2,16).

2. The dual of a translation hyperoval in PG(2,q) for any even q.

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2.2.4 Maximal arcs in small Desarguesian planes

1. The plane PG(2,8) has up to isomorphism only one maximal arc of degree 4; it is ofDenniston type and is the dual of the regular hyperoval.

2. The plane PG(2,16) has up to isomorphism two maximal arcs of degree 8: the dualof the regular hyperoval which is of Denniston type, and the dual of the Lunelli-Sce hyperoval which is of Mathon type. It has two non-isomorphic maximal arcs ofdegree 4, both of Denniston type and both self-dual.

3. The plane PG(2,32) has 6 non-isomorphic hyperovals and hence the same numberof maximal arcs of degree 16. As far as the maximal arcs of Denniston type areconcerned, there is one of degree 4, its dual of degree 8, and the dual of the regularhyperoval which is a maximal arc of degree 16. Mathon gives in his original paper[82] a construction of 3 maximal arcs of degree 8 (and hence of 3 maximal arcs ofdegree 4), which are not of Denniston type. It has been proved in [39] that thereare no other maximal arcs of Mathon type of degree 8 and moreover a geometricconstruction of these three maximal arcs of degree 8 of Mathon type has been given.

4. Mathon mentions in his paper [82] that, neglecting the hyperovals and their duals, theknown maximal arcs in the plane PG(2,64) are as follows.

• There are 94 non-isomorphic maximal arcs of degree 16 of Mathon type known,4 of them are of Denniston type. Hence, there are also 94 non-isomorphicmaximal arcs known of degree 4.

• There are 71 non-isomorphic maximal arcs of degree 8 known, 2 of them areof Thas type and are related to the Tits ovoid, and are self-dual, 7 of them areof Denniston type and are also self-dual, the others are all of pure Mathon type.Mathon found 31 of them by computer, none of them self-dual, yielding in total62 maximal arcs of degree 8 of pure Mathon type.

2.3 Hermitian curves and unitals

2.3.1 Definitions and constructions

A unital or Hermitian arc in any projective plane π of order q2 is a set U of q3 + 1 pointssuch that every line of the plane contains 1 or q + 1 points of U. Given a unital U and apoint P off U, there are q+1 tangent lines to U from P giving, as intersection with U, q+1points called the feet of P.

Although many examples of unitals in non-Desarguesian planes do exist (and even uni-tals as 2− (q3 + 1,q + 1,1) designs, non-embeddable in projective planes) the aim of thischapter is to investigate only unitals in Desarguesian planes. The classical example is theHermitian curve, also called the classical unital in PG(2,q2) that has as canonical equation

xq+10 + xq+1

1 + xq+12 = 0.

Let H be a Hermitian curve of PG(2,q2), then on every point of H there is a unique tangentline and for every point P off H , the feet are the q + 1 points of the Baer subline given by

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the intersection of the polar line of P with H . Every line which is not a tangent line meetsH in a Baer subline. Every unital in PG(2,4) is a Hermitian curve. However, for everyq > 2, there are unitals in PG(2,q2) that are not Hermitian curves.

Buekenhout [33] has constructed unitals in translation planes π of order q2 using theAndré-Bruck-Bose representation (see Section 2.2.2). He proved that if H is a classicalunital, then the corresponding set H ∗ in PG(4,q) is either a quadric Q(4,q) intersecting thespace Σ∞ in a regulus of the regular spread S (if `∞ is a secant of H ) or it is a quadratic conewith vertex a point V on a line t of the regular spread at infinity and base an elliptic quadricmeeting Σ∞ at a point of t \ {V} (if `∞ is a tangent line to H ). Conversely, if U∗ is anovoidal cone in PG(4,q) with base an ovoid O of a PG(3,q) meeting Σ∞ in a tangent planeto O, containing a line t of S and with vertex a point V on the line t such that U∗∩Σ∞ = t,then the corresponding set U of points in PG(2,q2) forms a unital which has the line atinfinity, say `∞, as a tangent line. Hence, the construction by Buekenhout gives new unitalsfor q = 22e+1,h > 1, by choosing O a Tits ovoid (see Section 3.1.3). Up to now, the onlyknown ovoids of PG(3,q) are the elliptic quadrics and for q = 22e+1 > 8 also the Tits ovoids.

R. Metz [84] proved that in PG(4,q)\Σ∞ a conic `∗ can be chosen such that ` is a set ofq + 1 collinear points of PG(2,q2) \ `∞ not corresponding to a Baer subline. Such a coniccan always be completed to an ovoid (which has to be an elliptic quadric, see Theorem 3.9)giving by Buekenhout’s construction a non-classical unital in the plane PG(2,q2).

Remarks

1. Note that if the spread S is not a regular spread, then Buekenhout’s constructionyields a unital in the translation plane π constructed from S .

2. In the next sections we will use the following standard terminology.

• A unital coming from Buekenhout’s construction using a quadric Q(4,q) willbe called a Buekenhout unital.

• A unital coming from Buekenhout’s construction using a cone with base anovoid is called an ovoidal Buekenhout-Metz unital; it is called an orthogonalBuekenhout-Metz unital if the base is an elliptic quadric. Hence, the classicalunital in PG(2,q2) is an orthogonal Buekenhout-Metz unital, but from R. Metz[84] it follows that there exist in this plane orthogonal Buekenhout-Metz unitalsthat are not classical. An ovoidal Buekenhout-Metz unital with base a Tits ovoidis also called a Buekenhout-Tits unital.

3. There are no other unitals embedded in PG(2,q2) known at this moment.

2.3.2 Characterization theorems

It is known that every Buekenhout unital in PG(2,q2) is classical (see e.g. [13]).For q ≤ 3 it is known that every unital embedded in PG(2,q2) is an orthogonal

Buekenhout-Metz unital. (See [89] for PG(2,9)).As all ovoids of PG(3,q), q odd, are elliptic quadrics (see Section 3.1.3), the orthog-

onal Buekenhout-Metz unitals are the only possible ovoidal Buekenhout-Metz unitals inPG(2,q2), q odd.

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One of the first results on unitals in projective planes is due to Tallini-Scafati [105], whoproved the following theorem.

Theorem 2.13 ( [105]). Let U be a unital in a projective plane π of order q2. The set oftangents to U forms a dual unital Ud in the dual plane πd .

It is well known that the dual of a classical unital is classical. The same result holdsfor both Buekenhout-Metz and Buekenhout-Tits unitals in PG(2,q2) [9], [50], [35]. Henceevery known unital embedded in PG(2,q2) is either an orthogonal Buekenhout-Metz unitalor a Buekenhout-Tits unital.

A special class of orthogonal Buekenhout-Metz unitals has been constructed byHirschfeld and Szonyi [73] as the union of a partial pencil of conics in PG(2,q2), q odd.These are the only unitals containing conics. They use coordinates to describe these uni-tals. Later, a similar description in coordinates has been given that we summarize in thefollowing theorem.

Theorem 2.14 ( [9], [50]). Let

Uα,β = {(x,αx2 +βxq+1 + r,1) : x ∈ GF(q2),r ∈ GF(q)}∪{(0,1,0)}

for some α,β ∈ GF(q2) such that d = (βq −β)2 − 4αq+1 is a non-square in GF(q) if q isodd, while Tr(αq+1/(βq +β)2) = 0, β /∈ GF(q), if q > 2 is even. Then Uα,β is an orthogonalBuekenhout-Metz unital in PG(2,q2).

Coordinates have also been given by Ebert [51] for Buekenhout-Tits unitals obtainingthe following theorem.

Theorem 2.15 ( [51]). Let q = 22e+1, e > 1, let {1,δ} be a basis of GF(q2) over GF(q) andlet σ be the automorphism of GF(q) defined by σ : x 7→ x2e+1

. Let

U = {(x0 + x1δ,y0 +(xσ+20 + xσ

1 + x0x1)δ,1) : x0,x1,y0 ∈ GF(q)}∪{(0,1,0)}.

Then U is a Buekenhout-Tits unital in PG(2,q2).

Recall that a blocking set in a projective plane πn of order n is a set of points meetingevery line and containing no line. It is minimal if no proper subset is again a blocking set.

Theorem 2.16 ( [29], [32]). Let B be a minimal blocking set in πn. Then

n+√

n+1 6 |B|6 n√

n+1.

Moreover, if n = q2 and |B| = n +√

n + 1, then B is a Baer subplane; if |B| = n√

n + 1,then B is a unital.

Many characterization theorems for the known unitals embedded in PG(2,q2) areknown. We will just recall a few ones.

Theorem 2.17 ( [105]). Let U be a unital in πq2 . For a point Pi on U, denote by `i thetangent line to U at Pi. Suppose that U is reciprocal, i.e,. for every P1,P2,P3,P4 ∈ U nothree collinear, `1∩`2 ∈ P3P4 implies `3∩`4 ∈ P1P2. Then U is the set of absolute points ofa polarity. Hence, if πq2 = PG(2,q2), then U is classical.

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Theorem 2.18 ( [80], [53]). If U is a unital in PG(2,q2) with q > 2 such that every secantline meets U in a Baer subline, then U is classical.

This result has been improved by Ball, Blokhuis and O’Keefe in case q is a prime p.

Theorem 2.19 ( [11]). In PG(2, p2), p prime, a unital such that p(p2−2) secant lines meetin a Baer subline is classical.

The following theorem is a strong characterization theorem of the Hermitian curve.

Theorem 2.20 ( [110]). If U is a unital of PG(2,q2) such that the tangents of U at collinearpoints of U are concurrent, then U is classical.

The proof of this theorem is based on Segre’s “Lemma of tangents”, and on theorem2.18. The hypothesis of the previous theorem has been weakened by the following theorem.

Theorem 2.21 ( [7]). Let U be a unital in PG(2,q2), q > 2. If there are two points P1,P2 ∈U with tangent lines `1, `2, respectively, such that for all points Q ∈ `1 \ {P1} and R ∈`2 \{P2}, the corresponding feet are collinear, then U is classical.

The following theorems are nice characterization theorems of a classical unital as anovoidal Buekenhout-Metz unital.

Theorem 2.22 ( [16]). Let U be an ovoidal Buekenhout-Metz unital with the line l∞ tangentat P∞ in PG(2,q2). If there is a secant line not through P∞ that intersects U in a Baersubline, then U is classical.

The next theorem by K. Metsch [83] embeds PG(4,q) in PG(4,q2). It is well knownthat the form of an elliptic quadric in PG(3,q) yields a hyperbolic quadric in PG(3,q2),and hence the elliptic cone of PG(4,q), used to construct the orthogonal Buekenhout-Metzunital, becomes a hyperbolic cone. As far as the spread S in PG(3,q) is concerned, thereexist exactly two disjoint lines L and L′ of PG(3,q2), missing PG(3,q), and conjugate underthe Baer involution of PG(3,q2) fixing PG(3,q), such that S is the set of lines (regarded aslines of PG(3,q2)) intersecting L and L′. These two lines are commonly called the generatorlines of the regular spread S .

Theorem 2.23 ( [83]). Let U be an orthogonal Buekenhout-Metz unital. If L is a generatorline of the regular spread S, then U is classical if and only if L lies on the hyperbolic conein PG(4,q2).

More characterization theorems of orthogonal Buekenhout-Metz unitals are known, seefor instance [90] and [109] where they use the method of the so-called field reduction, butit would bring us too far to give these theorems in detail.

In terms of algebraic curves, the following characterization theorem for classical unitalsis a nice one.

Theorem 2.24 ( [72]). If U is an algebraic curve of degree q + 1 with |U| > q3 + 1 andwithout a linear component in PG(2,q2), then U is a classical unital.

As far as characterization theorems of ovoidal Buekenhout-Metz unitals in terms ofintersections with lines, the following theorems are worthwhile to mention.

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Theorem 2.25 ( [79]). Let U be a unital in PG(2,q2), q > 2, and let ` be some tangent lineto U. If all Baer sublines having a point on `, intersect U in 0,1,2, or q+1 points, then Uis an ovoidal Buekenhout-Metz unital.

Theorem 2.26 ( [78]). Let U be a unital in PG(2,q2), q odd, and let ` be a tangent line toU at P. Then U is an ovoidal Buekenhout-Metz unital if and only if for any two lines `1 and`2 such that `1∩ `2 = P, there is a Baer subplane B having ` as a secant line and satisfyingB ∩U = (`1∩U)∪ (`2∩U).

These theorems have been improved as follows.

Theorem 2.27 ( [35], [91]). If U is a unital of PG(2,q2), q > 2, containing a point P suchthat each of the q2 secant lines through P meets U in a Baer subline, then U is an ovoidalBuekenhout-Metz unital.

The proof ( [35] covers the case q even and q = 3, while [91] covers q odd, q > 3) isa sequence of lemmas using the André-Bruck-Bose representation for PG(2,q2), carefullyanalyzing the set U∗ of points in PG(4,q) corresponding to the unital U.

Regarding the intersection between a unital and a Baer subplane, the following result isknown.

Theorem 2.28 ( [14], Lemma 2.9). Let H be a Hermitian curve and let B be a Baersubplane in PG(2,q2). Then H ∩B is a (possibly degenerate) conic of B . Hence |H ∩B| ∈{1,q+1,2q+1}.

The converse of this result is also valid as the following theorem proves.

Theorem 2.29 ( [15]). Let U be a unital in PG(2,q2). If every Baer subplane meets U in1, q+1 or 2q+1 points, then U is classical.

As a general result regarding the intersection between a unital and a Baer subplane, thefollowing theorem is worthwhile to mention.

Theorem 2.30 ( [31], [65]). Let π be a projective plane of order q2 containing a unital Uand a Baer subplane B and let b be the number of lines secant to B and tangent to U. Then

|B ∩U |= 2(q+1)−b.

And regarding the intersection of an ovoidal Buekenhout-Metz unital and a Baer sub-plane, the following result is known.

Theorem 2.31 ( [15]). Let U be a unital tangent to `∞ in a translation plane πq2 with kernelcontaining GF(q). If every Baer subplane secant to `∞ meets U in 1 (mod q) points, thenU is an ovoidal Buekenhout-Metz unital.

O’Nan showed in [87] that the classical unital does not contain a configuration of fourdistinct lines which meet in six points (called an O’Nan configuration).

Wilbrink proved the following theorem.

Theorem 2.32 ( [117]). Let U be a unital in PG(2,q2), q even. If

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• U contains no O’Nan configuration;

• for each secant ` of U, point X ∈U \ ` and secant m on X meeting ` in a point of U,it holds that if Y is a point of (m∩U)\{X}, then there exists a secant `′ distinct fromm on Y which meets each secant on X that meets ` in a point of U.

Then U is classical.

We conclude this section discussing the configurations arising as intersection of twoHermitian curves and with some group theoretical characterizations of the unitals embeddedin PG(2,q2). First a more general result.

Theorem 2.33 ( [19]). Let H be a Hermitian curve and let U be a unital of PG(2,q2),q = ph, p prime, h ≥ 1. Then |H ∩U| ≡ 1 (mod p).

Theorem 2.34 ( [77]). Let H and H ′ be two distinct Hermitian curves of PG(2,q2). Thenone of the following configurations occurs for H ∩H ′:

• a point; a Baer subline; a Kestenband (q2−q+1)-complete arc;

• a set of q2 +1 points on q Baer sublines with a point in common;

• a set of q2 +1 points on q−1 Baer sublines on lines with a point in common plus twoother points;

• a set of q2 +q+1 points on q+1 Baer sublines with a point in common;

• a set of (q+1)2 points on q+1 Baer sublines on lines with a point in common.

The point sets with q2 +q+1 and (q+1)2 points together with the point set with q2 +1points on q−1 Baer sublines plus two points have been studied in detail in [44]. By usingthe André-Bruck-Bose representation of PG(2,q2), it is proved that these sets correspondto the elliptic or hyperbolic quadrics of a hyperplane in PG(4,q) \Σ∞ and to a quadratic3-dimensional cone. The other point set with q2 + 1 points has been studied in [46]. For astudy of the groups stabilizing H ∩H ′, see Giuzzi [61].

On the groups of the known unitals embedded in PG(2,q2), we just recall that the groupof the classical unital is the unitary group PGU(3,q2), it has order q3(q3 + 1)(q2− 1) andacts as a 2-transitive group on the points of the Hermitian curve.

Some group theoretical characterizations of the classical unital are given in the follow-ing theorems.

Theorem 2.35 ( [18]). Let U be a unital in PG(2,q2). If the group G of collineations fixingU is transitive on secant lines to U and is generated by involutions, then U is classical.

Theorem 2.36 ( [37]). Let U be a unital in PG(2,q2) fixed by a Singer subgroup of orderq2−q+1 of PGL(3,q2). Then U is classical.

The groups of the other orthogonal Buekenhout-Metz unitals and of the Buekenhout-Tits unitals can be found in [14].

Some group theoretical characterizations of both orthogonal Buekenhout-Metz andBuekenhout-Tits unitals in PG(2,q2) are given in the following theorems.

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Theorem 2.37 ( [2]). Let U be an ovoidal Buekenhout-Metz unital in PG(2,q2). If thereis a cyclic group of collineations of order q2−1 fixing two points of U and stabilizing U,then U is classical.

Theorem 2.38 ( [4], [2]). If U is a unital in PG(2,q2) fixed by a subgroup G of PGL(3,q2)such that:

• there is a point P of U fixed by G;

• G has a normal subgroup acting transitively on U \{P};

• the stabilizer in G of a point Q ∈ U \{P} has a cyclic subgroup of order q−1.

Then U is an orthogonal Buekenhout-Metz unital.

Theorem 2.39 ( [5]). Let U be an ovoidal Buekenhout-Metz unital in PG(2,q2). If there isa point P ∈ U such that the stabilizer of U in PGL(3,q2) has a subgroup that acts sharplytransitive on U \{P}, then U is an orthogonal Buekenhout-Metz unital.

Theorem 2.40 ( [52]). Let U be a unital in PG(2,q2) which is fixed by a subgroup ofPGL(3,q2) which is a semidirect product of a group of order q3 and a group of order q−1.Then U is an orthogonal Buekenhout-Metz unital.

Very recently among these lines the following theorems have been proved.

Theorem 2.41 ( [48]). If G has a subgroup of elations with center A of order q and asubgroup fixing both A and B of order a divisor of q−1 greater than 2(

√q−1), then U is

an ovoidal Buekenhout-Metz unital with respect to A.

Theorem 2.42 ( [48]). Suppose that q is either odd or q ∈ {4,16}. A unital U in PG(2,q2)is an orthogonal Buekenhout-Metz unital if and only if there exist two distinct points A andB on U such that G has a subgroup of elations with center A of order q and a subgroupfixing both A and B of order a divisor of q−1 greater than 2(

√q−1).

2.4 Characterizing subplanes of PG(2,q)

In this section we discuss characterization theorems regarding Baer subplanes and sub-planes of smaller order of PG(2,q) and some related results.

Theorem 2.43 ( [104]). Let K be a set of type (1,k) in a projective plane πn of order n.Then n = q2,k = q+1 and K is either a Baer subplane or a unital.

As far as the embedding of Baer subplanes is concerned, a more general theorem isknown.

Theorem 2.44 ( [26]). Let πm be a subplane of order m of a projective plane πn of order n.Then either m2 = n, that is πm is a Baer subplane of πn, or m2 +m ≤ n.

In particular, in PG(2,q), q = ph, there is a subplane PG(2, pt) for any t|h.Regarding sets of class [0,1,n] in PG(2,q) the following results are known.

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Theorem 2.45 ( [75]). If K is a proper point set of type (0,1,n), n > 4, in PG(2,q), then|K |6 q

√q+1.

Theorem 2.46 ( [115]). If K is a proper point set of class [0,1,n] in PG(2,q) with n >√q+1, then K is one of the following:

• either one or n collinear points;

• a Baer subplane;

• a unital;

• a maximal arc.

What about the configuration arising from the intersection of two Baer subplanes orsubplanes of smaller order?

The study of the intersection of Baer subplanes in PG(2,q2) started in [36] and wascarried on in [22], [116]. In these papers many properties regarding the intersection of twoBaer subplanes of PG(2,q2) were found. The main result was that two Baer subplanes haveas many points as lines in common. Later, all possible intersection configurations of twoBaer subplanes in PG(2,q2) have been determined.

Theorem 2.47 ( [102], [103], [76]). In PG(2,q2) let πq and π′q be two distinct Baer sub-planes, then πq∩π′q is one of the following configurations:

• the empty set;

• a point; two points; three points forming a triangle;

• a Baer subline `0 plus possibly a point not on `0.

Moreover, all previous configurations occur.

Recently, the previous theorem has been generalized for subplanes of any order. Indeedthe following theorem has been achieved.

Theorem 2.48 ( [47]). In PG(2,q), q = ph, let π be a subplane of order pt and let π′ be asubplane of order pt ′ with t|h and t ′|h. Then π∩π′ is one of the following configurations:

• the empty set;

• a point; two points; three points forming a triangle;

• a subline `0 over the biggest common subfield of GF(pt) and GF(pt ′) plus possibly apoint not on `0.

Moreover, if t|t ′, then all previous configurations occur.

Observe that the configurations coming from the intersection of two subplanes ofPG(2,q),q = ph, are of class [0,1,2, ps +1], for some s|h.

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3 Classical sets with few intersection numbers in PG(n,q), n≥ 3

3.1 Quadrics and quasi-quadrics

3.1.1 Definitions

In this section we are only interested in quadrics with an irreducible quadratic form andwhich can not be described in fewer variables (also known as non-singular quadrics). So weneglect the cones with vertex an m-dimensional space projecting a quadric in an (n−m−1)-dimensional space skew to the vertex, as well as the quadrics which degenerate in the unionof subspaces. For more details on the general theory of quadrics we refer for instanceto [71].

The following projective classification is part of standard knowledge.

Theorem 3.1. If Q is a non-singular quadric, then it is of one of the following types.

• n = 2m, Q is called parabolic, also denoted by Q(2m,q) and the quadratic form isequivalent to the following canonical form

Q(x0,x1, . . . ,x2m) = x20 + x1x2 + · · ·+ x2m−1x2m.

• n = 2m−1, in which case there are two non-equivalent quadrics.

– The hyperbolic quadric Q+(2m− 1,q) with quadratic form equivalent to thefollowing canonical form

Q(x0,x1, . . . ,x2m−1) = x0x1 + x2x3 + · · ·+ x2m−2x2m−1.

– The elliptic quadric Q−(2m−1,q) with quadratic form equivalent to the follow-ing canonical form

Q(x0,x1, . . . ,x2m−1) = f (x0,x1)+ x2x3 + · · ·+ x2m−2x2m−1,

with f (x0,x1) an irreducible quadratic form over GF(q).

The subspaces on Q of maximal dimension (also called the projective dimension of thequadric) are called the generators of the quadric. It is commonly known that the projectivedimension of the parabolic quadric Q(2m,q) and of the hyperbolic quadric Q+(2m− 1,q)is m−1, while the one of the elliptic quadric Q−(2m−1,q) is m−2.

It is a standard exercise to count the number of points |Q | of a quadric Q .

|Q(2m,q)| =q2m−1q−1

,

|Q+(2m−1,q)| =(qm−1 +1)(qm−1)

q−1,

|Q−(2m−1,q)| =(qm−1−1)(qm +1)

q−1.

If q is odd, the set of points of the quadric Q in PG(n,q) can be regarded as the setof absolute points of an orthogonal polarity. If q is even and n is odd, the quadric definesa symplectic polarity, if q and n are both even then no polarity is defined and the tangenthyperplanes to Q(2m,q) all have a unique point in common, the nucleus of the parabolicquadric (generalizing the same property for conics in PG(2,2h)).

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3.1.2 Characterization theorems

The set of points of an elliptic quadric as well as of a hyperbolic quadric in PG(2m−1,q)is a two-character set with respect to hyperplanes.

Indeed let Q−(2m − 1,q) be an elliptic quadric, then its point set is a set K of(qm−1−1)(qm+1)

q−1 points such that a hyperplane either intersects the quadric in a parabolic

quadric, hence in q2m−2−1q−1 points, or is a tangent hyperplane in which case it intersects

Q−(2m− 1,q) in a cone with vertex the tangent point X projecting an elliptic quadricQ−(2m− 3,q), from which follows that such a tangent hyperplane intersects the ellipticquadric in q(qm−1+1)(qm−2−1)

q−1 +1 points.The same argument holds for the hyperbolic quadric Q+(2m− 1,q) with point set

K a set of (qm−1+1)(qm−1)q−1 points such that each hyperplane meets it in either q2m−2−1

q−1 orq(qm−2+1)(qm−1−1)

q−1 +1 points.Every set K being of the same type with respect to hyperplanes as the elliptic quadric

(respectively hyperbolic quadric) is called in [41] an elliptic quasi-quadric (respectivelyhyperbolic quasi-quadric) and is denoted by K − (respectively K +).

In that paper the authors construct elliptic and hyperbolic quasi-quadrics. We will givehere only one construction.

Let Qε(2m−1,q) be a non-degenerate quadric in PG(2m−1,q) (ε =− for the ellipticquadric and ε = + for the hyperbolic quadric), m > 2. Let X be a point of Qε(2m−1,q). Thetangent (polar) space X⊥ of X with respect to the quadratic form for Qε(2m−1,q) is thenof dimension 2m−2, and X⊥∩Qε(2m−1,q) is the cone XQε(2m−3,q) with vertex X andbase a non-degenerate quadric Qε(2m−3,q) in some subspace Σ2m−3 of X⊥ of dimension2m−3 disjoint from X .

Let Q′ be a (quasi-)quadric in Σ2m−3 with the same parameters as Qε(2m− 3,q). Wethen call the set (Qε(2m−1,q)−\XQε(2m−3,q))∪XQ′ a pivoted set of Qε(2m− 1,q)with respect to X . Note that the size of a pivoted set is the same as the size of Qε(2m−1,q).

Theorem 3.2 ( [41]). Every pivoted set with respect to a point of Q−(2m− 1,q) (respec-tively Q+(2m− 1,q)) is an elliptic (respectively hyperbolic) quasi-quadric with the samecharacters as those arising from Q−(2m−1,q) (respectively Q+(2m−1,q)).

Remarks

1. If q = 2, there are more possible constructions of elliptic and hyperbolic quasi-quadrics, see [40] and [41] for more details; in this case these quasi-quadrics give riseto other structures such as symmetric designs with the symmetric difference property,Reed-Muller codes and bent functions.

2. A quasi-quadric in PG(3,2) is a quadric. In PG(5,2) there are five projectively in-equivalent quasi-quadrics of elliptic type and seven of hyperbolic type, see [114] formore details.

3. There also exist parabolic quasi-quadrics that are not quadrics; see [41] for moreinformation.

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Note that the pivoting construction is breaking up the lines and subspaces on thequadric, moreover one can repeat pivoting as much as one wants, implying that the familyof quasi-quadrics is quite wild. However, the following theorems are worthwhile to mentionin this context.

Theorem 3.3 ( [57]). Let K be a set of points in PG(n,q), where n > 4 and |K | ≥ q3 +q2 + q + 1, such that K intersects all planes in 1, a, or b points, b ≥ 2q + 1, K intersectsall solids in c, c+q, or c+2q points, c ≤ q2 +1, and there exist solids intersecting K in cpoints and in c+q points; then K is a non-singular quadric of PG(4,q).

De Winter and Schillewaert proved the following results in the same style.

Theorem 3.4 ( [93]). 1. If a set K of points in PG(4,q) intersects all planes and allsolids in the same number of points as quadrics do, then K is a parabolic quadricQ(4,q).

2. If a set K of points in PG(n,q) intersects planes and solids in the same number ofpoints as a quadric of PG(n,q) does, then K is one of the following:

(a) the space PG(n,q),

(b) a hyperplane of PG(n,q),

(c) a quadric of PG(n,q),

(d) a cone with vertex an (n−3)-dimensional space and base an oval, q even,

(e) a cone with vertex an (n−4)-dimensional space and base an ovoid, q even.

Theorem 3.5 ( [42]). An elliptic quasi-quadric in PG(n,q), n ≥ 4, q > 2, or a hyperbolicquasi-quadric in PG(n,q), n ≥ 3, q > 2, such that it also has the same characters withrespect to codimension 2 spaces is a quadric.

Remark

In cryptography, one is studying for instance maximum non-linear functions. Geometri-cally, these functions correspond to quasi-quadrics, see for instance [70] for more details.

3.1.3 Ovoids and generalizations

The elliptic quadric Q−(3,q) is a set of q2 +1 points, no 3 on a line. Every set K in PG(n,q)with the property that no 3 points are on a line is called a k-cap, with |K | = k. If n = 2,a k-cap is a k-arc and this case we already have treated in the beginning of this chapter.Hence, from now on n ≥ 3. A line of PG(n,q) will be called external, tangent, or secantto a cap according to whether it contains zero, one, or two points of the cap. A k-cap ofmaximal size has been called an ovaloid by Segre [97] and if q > 2 the maximal size isindeed q2 + 1, which was first proved by Bose [21] for q odd, by Seiden [101] for q = 4,and by Qvist [92] for q > 2 and even. Note that if q = 2, the 8 points of an affine subspaceof PG(3,2) is a maximal set of points no 3 on a line. We will discard this case for the restof the section. If O is an ovaloid of PG(3,q), q > 2, then for every point P on O, the tangentlines through P are in a plane, the tangent plane, and hence there are q2 +1 tangent planes

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and the other q3 + q planes intersect O in an oval. Actually, Tits [111] defined an ovoid tobe a set O of points in a projective geometry (not required to be finite nor Desarguesian)such that for any point P ∈ O the union of all lines ` with `∩O = {P} is a hyperplane. InPG(n,q) an ovoid can only exist if n ≤ 3. It is immediate from the definition of an ovoidthat in PG(3,q) it has size q2 + 1. Thus an ovoid of PG(3,q) is an ovaloid for q > 2, andwe will use the term ovoid for the rest of this section.

The concept of an ovoid has been generalized in many ways; it would us bring too farto discuss also these generalizations. However, a set O of points (so not necessarily a cap)in PG(n,q), n ≥ 3, such that the union of the tangents (1-secants) at each point are in ahyperplane, is called a semi-ovoid. It has been proved by Thas [107] that no semi-ovoidsexist in PG(n,q), n > 3, and that it is an ovoid if n = 3.

One might wonder whether an ovoid of PG(3,q) is necessarily an elliptic quadric. Theanswer is affirmative if q is odd, as proved independently by Barlotti [12] and Panella [88].It is however not the case if q is even. In this case, there is an ovoid known which is notthe elliptic quadric; the so-called Tits ovoid that exists if q = 22e+1, e ≥ 1, and it has thefollowing canonical form

{(1,zu+ zσ+2 +uσ,z,u) : z,u ∈ GF(q)}∪{(0,1,0,0)},

with σ : x 7→ x2e+1. One of the motivations of the study of Tits into ovoids of PG(3,q) and

his construction of this ovoid in [112], is the fact that the full stabilizer in PGL(4,q) of theovoid is the simple group of Suzuki, for this reason the ovoid is sometimes also called theTits-Suzuki ovoid. Together with the elliptic quadric Q−(3,q) they are characterized by thefact that their full stabilizer acts doubly transitive on the ovoid [113].

While the non-tangent plane sections of an elliptic quadric all are conics, those of theTits ovoid all are translation ovals (i.e., ovals invariant under a group E of elations of orderq such that all the elations in E have a common axis) which are not conics.

No other ovoid is known and actually the ovoids in PG(2,2h) are classified for h ≤ 5.One of the big research issues of the last years is to find whether the elliptic quadric and theTits ovoid are the only ovoids in PG(3,q) or not.

There are very nice characterization theorems known for the elliptic quadric and theTits ovoid. We will give a few.

The following theorem is a theorem by Brown.

Theorem 3.6 ( [25]). An ovoid of PG(3,q) is stabilized by a central collineation if and onlyif it is an elliptic quadric.

As already mentioned, Barlotti characterized the elliptic quadrics as the ovoids inPG(3,q), q odd. Actually, he proved a more general theorem, not assuming q odd.

Theorem 3.7 ( [12]). If every non-tangent plane intersects the ovoid O of PG(3,q), q > 2,in a conic, then O is the elliptic quadric.

Segre improved this theorem in 1959.

Theorem 3.8 ( [97]). An ovoid of PG(3,q), q ≥ 8, which contains at least 12(q3−q2 +2q)

conics must be an elliptic quadric.

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However, also this result has been improved by Brown in 2000 (using the theorem ofBarlotti).

Theorem 3.9 ( [24]). An ovoid of PG(3,q), q even, such that there is a plane intersectingthe ovoid in a conic, is an elliptic quadric.

So, the question arises what can be said if one of the non-tangent planes of PG(3,q), qeven, intersects the ovoid in an oval which is not a conic. One of the theorems we want tomention in this context is the following one.

Theorem 3.10 ( [85], [86]). Suppose that O is an ovoid in PG(3,q), q even.

1. O has a pencil of translation ovals if and only if O is either an elliptic quadric or aTits ovoid.

2. If each non-tangent plane section is an oval contained in a translation hyperoval,then O is an elliptic quadric or a Tits ovoid.

Finally, here is a theorem which is in the same style as Theorem 3.9.

Theorem 3.11 ( [23]). Suppose that O is an ovoid of PG(3,q), q = 2h, h > 1. If there is aplane intersecting O in a pointed conic, then either q = 4 and O is an elliptic quadric, orq = 8 and O is the Tits ovoid.

3.2 Hermitian varieties

In this section we discuss characterization theorems regarding Hermitian varieties ofPG(n,q2). A Hermitian variety of PG(n,q2) is a set of type (0,1,q + 1,q2 + 1)1 and itis a two character set with respect to hyperplanes. Hermitian varieties have been character-ized by using their intersection numbers with lines. The following theorem is a combinationof papers by Tallini-Scafati [106], Hirschfeld and Thas [74] and Glynn [63].

Theorem 3.12 ( [106], [74], [63]). Let K be a non-singular point set of type (1,r,q2 +1)1in PG(n,q2), n > 4,q > 2, such that 3≤ r ≤ q2−1 and there is no plane π such that π∩Kis of type (r,q2 +1) in π. Then K is the point set of a Hermitian variety H(n,q2).

Another characterization used the intersection of a Hermitian variety with planes insteadof lines. But this time not just the intersection numbers are required but also the intersectionstructure.

Theorem 3.13 ( [54]). Let K be a point set of PG(n,q) such that every plane section is a(possibly degenerate) Hermitian curve. Then K is a Hermitian variety.

Recently, using the intersection numbers with more than one family of subspaces, thefollowing characterizations have been obtained:

Theorem 3.14 ( [94]). Let K be a non-singular point set of PG(n,q2),n > 4, having thesame intersection numbers with respect to planes and solids as H(n,q2). Then K is thepoint set of H(n,q2).

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We remark that for all characterizations in [63], [74], [94], [106], one might remove thenon-singularity hypotheses in order to include also the singular Hermitian varieties.

It is however impossible to characterize Hermitian varieties using just their intersectionnumbers with respect to hyperplanes since quasi-Hermitian varieties can be constructed inthe same way (using pivoting) as was done in an earlier section for quasi-quadrics. Askinghowever that the point set has also the same intersection numbers with respect to codimen-sion two subspaces the following characterization has been obtained.

Theorem 3.15 ( [42]). Let K be a point set of PG(n,q2),n > 3, having the same intersectionnumbers with respect to hyperplanes and codimension two subspaces as H(n,q2). Then Kis the point set of H(n,q2).

About the intersection of two Hermitian surfaces H and H ′ of PG(3,q2) we recallthat in [62] Giuzzi describes all possible intersection configurations H ∩H ′ under the hy-pothesis that the pencil generated by H and H ′ contains at least one degenerate Hermitiansurface (obtaining several possible intersection configurations).

In [98] B. Segre defines two Hermitian surfaces in PG(3,q2) to be permutable if andonly if their associated polarities u, respectively u′ commute and he proves the followingtheorem.

Theorem 3.16 ( [98]). If q is odd and H , H ′ are permutable Hermitian surfaces ofPG(3,q2), then uu′ is a projectivity with two skew lines of fixed points, called the fun-damental lines of H and H ′.

A point set of q2 + 1 mutually skew lines in PG(3,q2) with exactly two transversals iscalled a pseudo-regulus. This notion was introduced by J. Freeman in [60], where he provedthat any pseudo-regulus can be extended to a spread of PG(3,q2). The set of (q2 +1)2 pointscovered by a pseudo-regulus is called a hyperbolic QF -set in [45]. It is one of the possibleintersection configurations of two Hermitian surfaces. Indeed the following holds.

Theorem 3.17 ( [6]). Let H and H ′ be two permutable Hermitian surfaces of PG(3,q2),q odd. If the fundamental lines are contained in H ∩H ′, then H ∩H ′ is the point set of apseudoregulus.

The hypotheses in the previous theorem are weakened in [46].

Theorem 3.18 ( [46]). Let H and H ′ be two distinct Hermitian surfaces in PG(3,q2)with associated polarities u and u′, respectively. Suppose that L and M are two skew linescontained in B = H ∩H ′. Then B is a hyperbolic QF -set (point set of a pseudo-regulus)with transversals L and M if and only if u and u′ agree on the points of L∪M.

Finally the next theorem yields a complete classification for H ∩H ′.

Theorem 3.19 ( [49]). Let H and H ′ be two non-degenerate Hermitian surfaces inPG(3,q2) and let B = H ∩H ′. If the Hermitian pencil they generate contains only non-degenerate Hermitian surfaces, then one of the following four cases must occur:

• H contains exactly two skew lines and q4−1 other points;

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• H contains exactly two skew lines L and M, a third line N intersecting both L and M,and q4−q2 other points;

• H contains exactly four lines forming a quadrangle and q4−2q2 +1 other points;

• H is ruled by a pseudo-regulus.

Moreover, all these cases occur.

Note that the intersection of two Hermitian varieties in PG(n,q2) is always a set of class[0,1,2,q + 1,q2 + 1]1, but very little is known regarding the intersection of two Hermitianvarieties of PG(n,q2) for n > 4.

3.3 Subgeometries

In Theorem 2.46 we have described the classification of sets of class [0,1,r]1 in PG(2,q)under the condition that r ≥ √

q + 1. Actually, in the same paper Ueberberg has given,under the same condition, the classification of sets of class [0,1,r]1 in PG(n,q). He provedthe following theorem.

Theorem 3.20 ( [115]). If K is a proper point set of class [0,1,r]1 in PG(n,q) with r ≥√q + 1 and K spans the full space. Then it is a Baer subgeometry of PG(n,q) or an affine

subspace of PG(n,q).

The study of the intersection of two Baer subgeometries of PG(n,q2) has been carriedout in [30] and [102]. In these papers the authors prove that the number of common points oftwo Baer subgeometries of PG(n,q2) equals the number of common hyperplanes (see [17]for a generalization of this result to other subspaces different from hyperplanes). In [102]also all possible intersection configurations in PG(3,q2) have been conjectured. Finallyin [76] a complete determination of the structure of these intersections has been determined,solving Sved’s conjecture in the positive.

Theorem 3.21 ( [76]). Let B1, . . . ,Bk be Baer subgeometries of subspaces of PG(n,q2).The following statements are equivalent.

1. The Baer subgeometries B1, . . . ,Bk satisfy the following two conditions:

• k ≤ q+1

• 〈B1, . . . ,Bi−1,Bi+1, . . . ,Bk〉∩ 〈Bi〉= /0, for every i = 1, . . . ,k.

2. There exist two Baer subgeometries B and B ′ of PG(n,q2) such that B ∩B ′ = B1∪. . .∪Bk.

The previous theorem has been recently generalized by determining all possible inter-section configurations of any two subgeometries of PG(n,q).

Theorem 3.22 ( [47]). Let G and G ′ be two subgeometries of PG(n,q), q = ph, of order pt

and pt ′ respectively, with t ≤ t ′, and let m = gcd(t, t ′).If G ∩G ′ is non-empty, then G ∩G ′ = G1∪ . . .∪Gk, with k ≤ q−1

pt′−1and with G1, . . . ,Gk

subgeometries of order pm of independent subspaces of PG(n,q).

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In the same paper the authors prove also the vice versa of the last theorem under theassumption t|t ′.

Theorem 3.23 ( [47]). Let t and t ′ be two positive divisors of h with t|t ′. Let k ≤min{n+1, q−1

pt′−1} and let G1, . . . ,Gk be subgeometries of order pt of independent subspaces

of PG(n,q). Then there exist two subgeometries G and G ′ of order pt and pt ′ , respectively,of PG(n,q) such that G ∩G ′ = G1∪ . . .∪Gk.

Open problems

1. Is it possible to find all maximal arcs in PG(2,32)?

2. The dual of a Mathon maximal arc is again a Mathon maximal arc if and only if it isof Denniston type. There are three maximal arcs of degree 8 of Mathon type that arenot of Denniston type in PG(2,32), and hence 3 maximal arcs of degree 4 that arenot of Mathon type. Is it possible to give a direct construction of these maximal arcs?

3. What is the minimum number of secant lines being Baer sublines one needs to con-clude that a unital U is a Buekenhout-Metz unital ?

4. Determine all subsets of class [0,1,r] in PG(2,q) with r <√

q+1.

5. Remove the hypothesis t|t ′ in Theorem 2.48.

6. Let H be a point set of PG(n,q2) with the same number as the number of points ofa Hermitian variety and such that all hyperplane sections are (possibly degenerate)Hermitian varieties. Is it true that H is a Hermitian variety?

7. Determine all the possible intersections of two Hermitian varieties of PG(n,q2).

8. Determine all subsets of class [0,1,r]1 in PG(n,q), with r <√

q + 1, generatingPG(n,q).

9. Remove the hypothesis t|t ′ in Theorem 3.23.

10. Let A be a point of a unital U in PG(2,q2). Suppose there is a group of elations ofPG(2,q2) with center A (and axis the tangent line tA at A to U) stabilizing U. Is ittrue that U is a Buekenhout-Metz unital?

11. Are there other unitals, non isomorphic to Buekenhout-Metz unitals, in PG(2,q2)?

12. Suppose there is a point P of a unital U such that for every point of tP \{P} the feetare collinear. Is it true that U has to be an ovoidal Buekenhout-Metz unital?

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